Finally we would like to study how a free particle responds when there is exponential growth or decay of the flat background metric. We will study the motion of the test particle using geodesic acceleration similar to the case when one studies gravitational wave [1]}, [2]}. The explicit formula of geodesic acceleration is given by [3]},
\( A^{\rho } = \frac{D^2}{D\tau ^2} S^{\rho } = R^{\rho }_{\,\,\,\sigma \mu \nu } T^{\sigma }T^{\mu } S^{\nu }\,,\) 
